We are tasked with finding $f(g(x))$, where:

• $f(x) = x^2 - x - 1$
• $g(x) = -x - 4$

Step-by-step solution:

1. Substitute $g(x)$ into $f(x)$:

   Since $f(x) = x^2 - x - 1$, we replace every $x$ in $f(x)$ with $g(x) = -x - 4$.

   $f(g(x)) = (-x - 4)^2 - (-x - 4) - 1$

2. Simplify $(-x - 4)^2$:

   Use the formula $(a + b)^2 = a^2 + 2ab + b^2$ where $a = -x$ and $b = -4$.

   $(-x - 4)^2 = (-x)^2 + 2(-x)(-4) + (-4)^2$ $= x^2 + 8x + 16$

3. Substitute this into the equation:

   Now substitute the simplified form of $(-x - 4)^2$ into the expression for $f(g(x))$:

   $f(g(x)) = (x^2 + 8x + 16) - (-x - 4) - 1$

4. Simplify further:

   Distribute the minus sign in the second term:

   $f(g(x)) = x^2 + 8x + 16 + x + 4 - 1$

5. Combine like terms:

   $f(g(x)) = x^2 + 9x + 19$

Final answer:

$f(g(x)) = x^2 + 9x + 19$

Let me know if you need further details or clarification.

Here are 5 related questions for you to explore:

1. What is $g(f(x))$ given the same functions?
2. How do we find the inverse of $f(x)$?
3. What is the vertex of the quadratic $f(x) = x^2 - x - 1$?
4. How do you find the domain and range of $f(g(x))$?
5. How do you graph the composition $f(g(x))$?

Tip: When dealing with composition of functions, always start by substituting the inner function into the outer function's formula and then simplify step by step!